Application of Linear Programming to the Theory of the Firm: Including an Analysis of Monopolistic Firms by Non-Linear Programming

By Robert Dorfman

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1951.

• Language: English
• Publisher: University of California Press
• Release date: Nov 15, 2023
• ISBN: 9780520339446

Robert Dorfman

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PUBLICATIONS OF THE

BUREAU OF BUSINESS AND ECONOMIC RESEARCH

Previously published in this series:

A TREATISE ON WAR INFLATION by William Fellner (1942)

BREAD AND DEMOCRACY IN GERMANY by Alexander Gerschenkron (1943)

THE ECONOMICS OF THE PACIFIC COAST PETROLEUM INDUSTRY.

PART I: MARKET STRUCTURE by Joe S. Bain (1944)

LAND TENURE PROBLEMS IN THE SANTA FE RAILROAD GRANT AREA by Sanford A. Mosk (1944)

NATIONAL POWER AND THE STRUCTURE OF FOREIGN TRADE

by Albert O. Hirschmann (1945)

THE ECONOMICS OF THE PACIFIC COAST PETROLEUM INDUSTRY. PART 2: PRICE BEHAVIOR AND COMPETITION by Joe S. Bain (1945)

CALIFORNIA BUSINESS CYCLES by Frank L. Kidner (1946)

MONETARY POLICIES AND FULL EMPLOYMENT

by William Fellner (1946, 1947)

THE ECONOMICS OF THE PACIFIC COAST PETROLEUM INDUSTRY. PART 3: PUBLIC POLICY TOWARD COMPETITION AND PRICING by Joe S. Bain (1947)

THE STRUCTURE OF TRANSCONTINENTAL RAILROAD RATES

by Stuart Daggett and John F. Carter (1947)

THE CHANGING COMPETITIVE STRUCTURE OF THE WHOLESALE GROCERY TRADE:

A CASE STUDY OF THE LOS ANGELES MARKET by Ralph Cassady, Jr., and Wylie L. Jones (1949)

TAXING MUNICIPAL BOND INCOME by Lyle C. Fitch (1950)

CRISIS IN BRITAIN: PLANS AND ACHIEVEMENTS OF THE LABOUR GOVERNMENT by Robert A. Brady (1950)

SCIENTIFIC METHOD FOR AUDITING. APPLICATIONS OF STATISTICAL SAMPLING THEORY TO AUDITING PROCEDURE by Lawrence L. Vance

A COMPREHENSIVE CLASSIFIED MARKETING BIBLIOGRAPHY. PARTS I AND II by David A. Revzan (1951) application of linear programming to the theory of the firm

APPLICATION OF LINEAR PROGRAMMING TO THE THEORY OF THE FIRM

INCLUDING AN ANALYSIS OF MONOPOLISTIC FIRMS BY NON-LINEAR PROGRAMMING

ROBERT DORFMAN

A PUBLICATION OF THE BUREAU OF BUSINESS AND ECONOMIC RESEARCH, UNIVERSITY OF CALIFORNIA

PUBLISHED BY THE UNIVERSITY OF CALIFORNIA PRESS BERKELEY AND LOS ANGELES NINETEEN FIFTY ONE

University of California Press Berkeley and Los Angeles, California Cambridge University Press, London, England

Copyright 1951

By the Regents of the University of California Printed in the United States of America

BUREAU OF BUSINESS AND ECONOMIC RESEARCH

William Crum, Chairman

Robert A. Brady Charles A. Gulick Roy W. Jastram Maurice Moonitz

Sanford A. Mosk, Acting Director

The opinions expressed in this study are those of the author. The functions of the Bureau of Business and Economic Research are confined to facilitating the prosecution of independent scholarly research by members of the faculty.

Preface

This monograph, except for the last part of chapter iii,was prepared as a doctoral dissertation in the Department of Economics, University of California. The committee in charge of the research was headed successively by Professors William Fellner and R. A. Gordon. Chapter iv, especially, bears the imprint of Professor Gordon’s strict and constructive criticisms. Another member of the committee, Professor G. C. Evans of the Department of Mathematics, served as the guardian of the author’s mathematical conscience. Whatever mathematical elegance the treatment contains must be credited to the high standards which Professor Evans imposed and insisted on.

All work in the field of linear programming must be deeply indebted to Dr. George B. Dantzig of the Department of the Air Force, who opened up the field and has been one of its most productive investigators. But Dr. Dantzig’s contribution to the present monograph goes far beyond that. He gave much of his time to instructing the author in the concepts and methods of linear programming and after the research was under way allowed his patience to be imposed on many times when difficulties beyond the mathematical competence of the author threatened to bring the project to an ignominious halt. Professor A. W. Tucker, Princeton University, and Professor E. W. Barankin, University of California, were also consulted about mathematical difficulties, and the author is gratified to note that each of them subsequently undertook mathematical studies of first importance in problems suggested by this thesis. Professor Tucker, especially, by his pioneering study of Non-Linear Programming, cleared the way for the treatment of quadratic programming, which appears in chapter iii.

If this volume were dedicated to anyone, it would be to the officers and officials who staff the headquarters of the United States Air Force. These far-sighted men, faced with the difficulties of administering one of the largest integrated organizations in the world, have had the vision required to turn away from traditional methods of administration and to encourage an ambitious program of research into the theory of programming. The research whose results are reported in this monograph was originally undertaken as part of that research program and was inspired to a great extent by the needs of the Air Force. It is the author’s earnest wish that these results may assist in some small measure in easing and improving the work of Air Force headquarters. But, of course, the author also feels that the theories here developed invite far wider application.

I wish to thank the various publishers who have given permission for the use of quotations from their publications, including:

Harper & Brothers for permission to quote from Kenneth E. Boulding, Economic Analysis, 1941.

Harvard University Press for permission to quote from Paul A. Samuelson, Foundations of Economic Analysis, 1948.

Houghton Mifflin Company for permission to quote from Frank H. Knight, Risk, Uncertainty and Profit, 1921.

The Macmillan Company for permission to quote from Joan Robinson, The Economics of Imperfect Competition, 1948.

Oxford University Press, Inc., for permission to quote from J. R. Hicks, Value and Capital, 1941.

University of California R. D.

Berkeley, California

April 13, 1951

Contents 1

Contents 1

CHAPTER I Two Approaches to the Theory of the Firm

1. Historical Perspective.

2. The Marginal Analysis

3. Basic Concepts of Linear Programming

CHAPTER II The Competitive Firm Using Fixed Factors

1. Formulation of the Problem

2. The Basic Theorem

3. Computational Procedure

4. The Three Assumptions

5. Firm with Two Fixed Factors

6. Price Imputation for Fixed Factors

7. The Concept of the Dual

CHAPTER III Production Scheduling for Monopolized Products

1. The Monopolistes Problem

2. The Optimum Program

3. Application to Monopsony

4. Quadratic Programming

CHAPTER IV Assumptions, Limitations, and Possibilities

1. General Postulates of Maximization

2. The Specific Postulates of Linear Programming

3. The Problem of Time

4. Variations of Linear Programming

5. What is Linear Programming

References

CHAPTER I

Two Approaches to the

Theory

of the Firm

1. Historical Perspective.

The modern theory of the firm as expounded by Chamberlin (3)1 , Hicks (16), Robinson (31), Samuelson (32), Viner (38), et al., traces its lineage back to Ricardo’s discussion of the problems of English agriculture a century and a half ago. Ricardo and his contemporary, Malthus, share credit for the development of the concept of deceasing returns, the progenitor of the modern U- shaped cost curve. The classicists also used the idea of incremental variation of factors of production and thus set the stage for the modern marginal analysis. The fact that an analytic apparatus which was inspired by the predicament of early nineteenth-century English agriculture is so deeply embedded in the analysis of modern industrial problems calls for a reconsideration of the appropriateness of the model.

Ricardo’s theory (30) was based on a highly abstracted model of a single type of farming: English wheat cultivation. He grouped all things necessary for the cultivation of wheat into three factors of production: land, labor, and capital. He assumed that these factors of production could be combined in, essentially, any proportions desired. The technical process involved was considered from a broad point of view which ignored the existence of subsidiary steps and intermediate products; the three necessary factors were applied and after a proper interval the final product emerged. The entire process resulted in a single, homogeneous commodity —wheat.

Ricardo applied this model, essentially, to calculate the distribution of product between the landowner, who supplied the land, and the farmer, who provided the capital. He assumed that the total quantity of land was fixed and this total